Newton Polyhedra and an Oscillation Index of Oscillatory Integrals with Convex Phases

  • Isroil A. Ikromov
  • Akhmadjon Soleev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4194)


In this paper we obtain an analog of Schultz decomposition for arbitrary convex smooth functions. We prove existence of adapted coordinate systems for analytic convex functions. We show that the oscillation index of oscillatory integrals with analytic phases is defined by the distance between Newton polyhedron constructed in adapted coordinate systems and the origin.


Convex Function Polynomial Function Phase Function Remainder Term Oscillatory Integral 
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  1. 1.
    Arnol’d, V.I.: Remarks on the method of stationary phase and Coxeter numbers. Uspechi Mat. Nauk (in Russian) 28(5), 17–44 (1973); English. transl. Russ. Math. Surv. 28(5), 19–58 (1973)Google Scholar
  2. 2.
    Atiyah, M.F.: Resolution of singularities and division of distributions. Comm. Pure Appl. Math. 23(2), 145–150 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arnol’d, V.I., Gusein-zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps. In: Monodromy and Asymptotic of Integrals (in Russian), vol. II. Nauka, Moscow (1984), English. transl. Birhäuser (1988)Google Scholar
  4. 4.
    Bernstein, I.N., Gel’fand, I.M.: Meromorphic property of the functions p λ. Funct. Anal. Appl. 3(1), 68–69 (1969)Google Scholar
  5. 5.
    Karpushkin, V.N.: Theorems on uniform estimates of oscillatory integrals with phases depending on two variables. In: Proc. Sem. I.G. Petrovskii, vol. 10, pp. 50–68 (1984)Google Scholar
  6. 6.
    Varchenko, A.N.: Newton polyhedrons and estimates of oscillating integrals. Funct. anal. and appl. 10(3), 175–196 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Soleev, A., Aranson, A.: Calculation of a polyhedron and normal cones of its faces. In: Preprint Inst. Appl. Math. Russian Acad. Sci., Moscow, vol. (36) (1994)Google Scholar
  8. 8.
    Schultz, H.: Convex hypersurface of finite type and the asymptotics of their Fourier transforms. Indiana University Math. Journal 40(3), 1267–1275 (1991)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Isroil A. Ikromov
    • 1
  • Akhmadjon Soleev
    • 1
  1. 1.Department of MathematicsSamarkand State UniversityUzbekistan

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